Adding to what has been said here (and it's all been good!)

DnD has a history and thus there are layers to the paladin. To understand it you need to understand what it was and not just what it is now.

In 1st edition the paladin was arguably the strongest class (assuming the paladin had a holy avenger). It was balanced by way of some pretty strong restrictions on role play and some hefty requirements re: ability score. And given that the default world was Greyhawk and not Faerun playing lawful good was a challenge since evil is more powerful than good in Greyhawk.

Paladins had a 10 foot circle of dispel magic surrounding them and protection from evil field that went from 10 to 30 feet. In 1st protection from evil prevented demons, devils, elementals and summoned creatures from directly striking someone within the area and ranged attacks had -2 penalty. When a spell made it through the dispel zone the paladin still got a +2 to savings throws. And the paladin's sword was +5 weapon meaning it could be used against major deities. A paladin had a reasonable chance to solo one of the demon princes. And the list of abilities goes on: https://tablo.com/david-thomson/1st-ed-ad-and-d-the-paladin

But the Paladin had to keep a charisma of 17 -- which was the default dump stat of 1st -- and they took a vow of poverty that required that they give away all gold at the end of adventure. This also meant couldn't own more than 5 items. If the paladin committed an evil act they would fall and become an inferior form of fighter -- which was a form of insta death for the character because the toon was no longer worth playing. So upholding one's lawful good oath was a constant worry on the part of the human playing a paladin. The discussion of whether a paladin was culpable for acts taken while being charmed or possessed was a frequent point of discussion.

And that's still the paladin people think of when they use the word -- the archetype of the lawful good crusader. The avatar from the ultima series, Joan of Arc, El cid, Lancelot, etc.

2nd edition moved away from insta death like mechanics and (eventually) gave mechanisms for the paladin to atone for evil actions and loosened the poverty requirements. Now, 2nd ed was hot mess and you will some sources that say that fallen paladins are always fallen and others say the atonement spell and a quest will restore a paladin. BG2 had a redemption option for fallen paladins -- TSR / WotC was okay with it redemption became official.

https://adnd2e.fandom.com/wiki/Paladin

As other have said the character of Lord Soth captured the imagination of many and that gave rise to the blackguard class. Instead of being a weaker form of fighter a fallen paladin could become a champion of evil. In 3rd edition allowed paladins and blackguards with paladins still being lawful good but able to atone. Paladins who didn't atone had to choose a second class to progress. Which makes becoming a blackguard an obvious temptation -- imagine regaining all the power you lost . . .

In 5th you can be any alignment so it's sometimes difficult to disentangle the archetype from it's current manifestation.


Question 1.

In 1 - 3 eds the answer would have been yes. Paladins would have been part of some rule-bound, hierarchical order. Now, as Niara says, it's up the DM.

Question 3. I think this is where the history comes in. 5e allows DMs to use concepts from earlier editions (and the first modules actually had suggestions on how to run them using earlier rulesets). So the short answer is "up to the DM" the long answer is depends on how much the player and DM are drawing on the archetype that 1st edition created. If the player doesn't care for the oath but just wants to goodies that come with the class the DM can decide that the player isn't really taking the role playing requirements for the class seriously and punish the player.

Question 4. Of course up to players and DM but I'd say "smite the oathbreaker"!

Question 5. I could imagine a worshiper of Eldath serving as a "peacekeeper", smiting all those who break the peace. Were I DM I would expect the player to always try to avoid violence and to minimize the impact. Could be done in all seriousness or with the same humor as the old restrictions on clerics using bladed weapons -- "we're sworn not to draw blood, so we bludgeon people instead"!